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The University of Mumbai is one of the oldest and premier Universities in India. Around 550 affiliated colleges and 55 Departments runs under University of Mumbai.
Here is the list of MCA colleges affiliated to University of Mumbai
1. Veermata Jijabai Technological Institute (VJTI)
2. Late Bhausaheb Hiray S. S. Trust's Institute of Computer Application
3. Mumbai Educational Trust's Institute of Computer Science
4. S.I.E.S. College of Management Studies
5. Thakur Educational Trust's (Regd.) Thakur Institute of Management
6. Bharati Vidyapeeth's Institute of Management & Information Technology
7. National Centre for Rural Development Sterling Institute of Technology & Management
8. Navyug Vidyapeeth Trust's College of Master of Computer Application
9. Mandar Education Society's Rajaram Shinde College of MCA
10. Finolex Academy of Management & Technology
11. Vivekanand Education Society's Institute of Technology
Eligibility: you should be a graduate in any faculty of a statutory University with minimum 45% marks (minimum 40% for reserved category) and Mathematics at XII or higher level. Final Year Degree Students will be allowed to appear at the Entrance Test. There will be an aptitude test for the admission in MCA course. MCA Provisional selection will be based on the Entrance Test conducted by IDE, University of Mumbai.
Contact every college individually for more information regarding admission procedure. You can find more colleges affiliated to University of Mumbai for MCA by clicking here.
You can compare colleges as per your preference by using compare colleges tool. By using this tool you can compare 1 to 4 colleges at once.

To be eligible to get admission in any of the decent Management Institute a student needs to have a graduation degree from a recognized University with 50% and above aggregate marks. For doing MBA you have to appear for entrance exams such as CAT or CMAT. After that you have to go through Group Discussion and Personal Interview round and according to your results you will be eligible for specific colleges.
I would like to add here few of the things which you must keep in mind are: First and foremost the recognition of the Institute (the institute must be recognized by UGC or AICTE), the course curriculum and the study material, infrastructure, whether there is faculty or online support or not, last but not least Institutes reputation in the job market for which you should visit the institute.